Supplementary Information: Three-dimensional quantum photonic elements based on single nitrogen vacancy-centres in laser-written microstructures

Transcription

1 Supplementary Information: Three-dimensional quantum photonic elements based on single nitrogen vacancy-centres in laser-written microstructures Andreas W. Schell, 1, a) Johannes Kaschke, 2 Joachim Fischer, 2 Rico Henze, 1 Janik Wolters, 1 Martin Wegener, 2 and Oliver Benson 1 1) Nano-Optics, Institute of Physics, Humboldt-Universität zu Berlin, Newtonstraße 15, D Berlin, Germany 2) Institute of Applied Physics, DFG-Center for Functional Nanostructures, Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany a) Electronic mail: andreas.schell@physik.hu-berlin.de 1

2 SPECTRAL MODE ANALYSIS For a quantitative analysis of the spectrum measured from the background fluorescence by a grating spectrometer (see Figure 1) we performed a segmented Fourier transformation on the data. We divided the set into 25 nm wide segments and performed numerical fast Fourier transformations on each of them individually. Afterwards we matched the frequency amplitude ranges by normalizing the different segments to each other, colour coded the resulting profiles, and merged the different wavelength ranges to obtain the plot shown in Figure 2. The different Fourier components of the resonator s whispering gallery modes (WGMs) are clearly visible as bands evenly spaced by an integer multiple of the lowest observable base frequency. This frequency component is directly related to 1/FSR and corresponds well with the theoretical estimated value based on the size of the discs. The black dotted lines are fits to this value and its different higher-order multiples. FIG. 1. Spectrum of the background fluorescence when excited through one waveguide end and collected on the other (see main text) 2

3 FIG. 2. Normalised segmented Fourier transform showing the mode structure of the resonator s WGMs. The black dotted lines represent the different Fourier components of the collected spectrum. They are fits to the expected Fourier frequencies of the resonator s modes and connected to the wavelength dependent FSR of the system. The lowest frequency component directly relates to 1/FSR. Within the determined spectral window a constant refractive index of around 1.5 was assumed for resonators with 20 µm diameter. 3

4 TUNING OF THE RESONATORS Intensity (arb.u.) Wavelength (nm) FIG. 3. Permanent tuning of a resonator under vacuum Figure 3 shows the tuning of a resonator under vacuum conditions. To tune the resonator, a 405 nm laser was focussed on its rim. The spectra shown were acquired from bottom to top in equal timesteps of 30 s. After some steps the mode begins to shift. Laser induced shifts of the resonance of resonators were previously observed by us 1 and others 2. Its physical or chemical origin is not understood yet, but it is assumed to be caused by a modification of the resonator s local shape or a change of the index of refraction of the resonators material. It is worth noting that tuning of the resonator is possible even without oxygen present. This is a very important feature, since it enables tuning of the resonator s whispering gallery modes to the zero phonon lines of the nitrogen vacancy centres. Since no oxygen is needed for this procedure, it can be used in a cryostat at low temperatures. In addition to the possibility of tuning the resonances to lines of the optical emitters, it would be also possible to match resonances of different resonators and achieve controlled coupling among them. 4

5 FIG. 4. Data used for the background correction. a and b are coincidence counts measured with pulsed excitation at the location of an NV centre at the resonator s rim (in a)) and at a location without an NV center next to it also on the rim (in b)), respectively. Both were normalised to an average of one at long times >1.3 µs over 256 periods). A second axis indicates the total counts per bin prior to normalisation. A binsize of 1.28 ns was used. The dotted lines show the bins, as they were used to obtain c. c shows the resulting antibunching with a g (2) (0) = 0.18 ± BACKGROUND CORRECTION In the case of continuous excitation of an emitter with intensity < n a order correlation functions g (2) a the intensity < n b >= b with g (2) b by: >= a, second and the quantum mechanically uncorrelated background of = 1, the joint second order correlation function is given ab = a2 g a (2) + b 2 + 2ab. (1) (a + b) 2 g (2) By knowing the intensities a and b it one can calculate the g (2) a a from a measurement of the joint correlation function g (2) ab. function of the bare emitter In case of pulsed excitation this background correction is more complicated, since the emission of emitter and background are correlated by the excitation laser. This problem can be avoided by only looking at time intervals corresponding to one laser period. In the analysis of an experiment s g (2) -data acquired with a Hanbury Brown and Twiss setup, this can be achieved by binning together all events in the time windows T max min the laser repetition time T (= 25 ns in our experiment) and n being an integer. T = n T ±, with 2 Figure 4 a and b show the normalised coincidence counts measured with pulsed excitation at the location of an NV centre at the resonator s rim (in a)) and at a location without 5

6 an NV center next to it also on the rim (in b)), respectively. Thus the data in figure 4a) corresponds to coincidences from signal and background whereas figure 4b) represents the background coincidences only. Binning to intervals of 1.28 ns was performed for better visibility. Normalisation was done by averaging for long times (>1.3 µs over 256 periods), where contributions from the NV centre s bunching behaviour are negligible. Also the photons normally missed in an HBT like start-stop configuration were taken into account. A comparison of the background photons rate with the intensity of the combined signal gave a = 1 3 and b = 2, equivalent to a background to signal ratio of 2. Figure 4 c shows the 3 resulting corrected g (2) (τ) function, with bins sizes corresponding to the laser repetition time. For zero time delay g (2) (0) = The error bar for this value stems from three contribution, with the main contribution stemming from the uncertainties in the ratio b a, which we assumed to be 10 % ( b a = 2.0 ± 0.2). This error is due to the fact that background and signal were measured at neighboring positions on the resonator s rim, but not at the same one (see Figure 5 b of the main text, ( a b ) = 0.20). The other two contributions are the statistical variations in the photon number for the zero time delay bin, which were both assumed to be Possonian ( (a) = 0.01 and (b) = 0.07). This results in an error of (g (2) (0)) = 0.21 after adding the contributions via Gaussian error propagation. REFERENCES 1 Wolters, J. et al. Enhancement of the zero phonon line emission from a single nitrogen vacancy center in a nanodiamond via coupling to a photonic crystal cavity. Appl. Phys. Lett. 97, (2010). 2 Lee, H. S. et al. Local tuning of photonic crystal nanocavity modes by laser-assisted oxidation. Appl. Phys. Lett. 95, (2009). 6